You can fly Qantas JFK-LAX-SYD, it's a popular route. I'd also recommend checking out Delta if you're flying economy. Emirates generally won't sell (for cheap) NYC-DXB-SYD, it's a longer way around the world. But honestly flying economy is very similar no matter which airline you are flying.

You also shouldn't lump together all Asian airlines, because many airlines are excellent in contrast to average/mediocre mainland Chinese airlines: e.g. ANA, JAL, Korean Air, EVA, China Airlines (from Taiwan), Cathay Pacific are all world-class.